Concepts

Gravity is an extraordinary force and understanding its more profound implications for the cosmological many-body problem requires many strategies. So far, we have followed two broad avenues of insight into the instability and clustering of infinite gravitating systems: linear kinetic theory and numerical N-body simulations. Now we turn onto a third avenue: thermodynamics. Classical thermodynamics is a theory of great scope and generality. It survived the relativity and quantum mechanical revolutions of physics nearly intact. In part, this was because among all theories of physics thermodynamics has the least physical content. Its statements relate very general quantities that must be defined anew, through equations of state, for each specific application. With this view, it is natural to ask whether thermodynamics also subsumes gravitating systems.

The answer is yes, with certain caveats and qualifications. Results of gravitational thermodynamics – gravithermodynamics, or GTD for short – are often surprising and counterintuitive compared to the thermodynamics of ordinary gases. Specific heats, for example, can be negative and equilibrium is a more distant ideal. Basically, these differences are caused by the long-range, unsaturated (unshielded) nature of gravitational forces. As a result, rigorous understanding of GTD is less certain than for ordinary thermodynamics. The present situation is a bit similar to the early thermodynamic gropings of Watt, Carnot, Kelvin, and Joule.

For a decade now, the Instituto de Astrofisica de Canrias (IAC) has hosted the Canary Islands Winter School of Astrophysics in which young astrophysicists from all over the world have the opportunity of meeting accredited specialists to study the topics of most active concern in present-day astronomy. During these ten years 80 lecturers and more than 600 students have attended the Winter School, an even higher number not being able to come due to the limited number of places available.

The X Canary Islands Winter School on Astrophysics was dedicated to Globular Clusters, one of the basic sources of our knowledge concerning the lives of the stars and the physics of their evolution.

The School intended to portray a thorough review of research in this field, covering all the relevant disciplines with the aid of the best possible international team of specialists (Canada, Italy, South Africa, Spain, the United Kingdom and the United States), including the theoretical and observational aspects of stellar populations, stellar evolution and chemical abundances, dynamics, variable stars, X-ray sources and the globular clusters of other galaxies.

We take the opportunity to thank local Canarian authorities - Cabildo Insular de La Palma, and Cabildo Insular de Tenerife, as well as the Town Hall of La Laguna, for their continuous support during this and also previous editions of the School.

This tenth Winter School marks a milestone on a long but gratifying journey, in spite of occasional difficulties.

Many motives spur astronomers toward numerical simulations. These computer experiments test well-defined theories, display outcomes of complex interactions, elicit quantitative comparisons with observations, and provoke new insights. Moreover, they are almost always guaranteed to lead to a publishable result. What could be finer and more delightful!?

Streams of simulations have therefore poured forth in abundance. They differ mainly in their assumptions about the amount, nature, and role of dark matter, and in their initial conditions. Most agree with some aspects of observations, but none with all. None, so far, are generally accepted as complete descriptions of galaxy clustering.

As computing power expands, each new generation essentially repeats these simulations with more complicated physical interactions, greater detail, higher resolution, and added parameters. While this development continues, it seems to me wiser not to discuss the latest examples here, for they will soon be as obsolete as their predecessors. Instead, we concentrate on the simplest case: the cosmological many-body problem. Even this reveals a richness of behavior that surpasses current understanding. Understanding is more than simulation, for it embeds the simulations in a much richer conceptual context.

Suppose that cosmological many-body clustering runs on forever. What will happen in the infinite future?

Standard Einstein–Friedmann universes suggest three main possibilities. If the Universe is closed (Ω0 > 1, k = +1) and recollapses into a singularity, all large-scale structure will eventually be destroyed in the big crunch. Whether anything can be resurrected from surviving seeds if the crunch is incomplete (Saslaw, 1991) is unknown. Oscillating universes are possible, though in practice we do not know if the physical requirements for repeated oscillations are consistent with reasonable equations of state. Oscillations whose amplitudes were too small to produce equilibrium would accumulate the debris of previous cycles. Quite apart from the question of increasing entropy, such models would probably require especially fine tuning to produce our observable Universe.

If the Universe is open and expands forever with negative curvature (Ω0 < 1, k = –1), it will expand so rapidly after redshifts z ≲ Ω0–1 (see 30.13) that new larger structures will generally cease to form, and the largest scale patterns at z ≈ Ω0–1 will be essentially frozen. These patterns then tend to expand homologously, becoming increasingly stretched and dilute in physical space: Pascal's nightmare. In models with a cosmological constant, the expansion may pause. But it will have to be carefully tuned, so the quasi-stationary period does not produce overclustering, and also satisfy other constraints.

The study of globular clusters has been and still is essential for furthering our knowledge of such astrophysical phenomena as stellar and galactic evolution, variable and X-ray emission stars, chemical abundances (primordial nucleosynthesis), etc. Globular clusters are ideal laboratories for testing theories of stellar evolution, the chemical evolution of the Universe and the dynamics of N-body systems. They are the oldest known objects whose ages can be independently determined, the closest in proximity to the origin of the Universe and the sole surviving structures of the first stages in the formation of the Galaxy. They provide us with important evidence concerning on the age and formation processes of the Galaxy. Globular Clusters are a fundamental unit of the known Universe, they are also found in all other galaxies within our observational grasp. They are possibly a necessary stage in the formation of galaxies.

Research on Globular Clusters covers a vast amount of territory that was reviewed and collected in the present book. From the photographic plate to the HST most recent results, the field of Globular Clusters was actualised and presented by Ivan R. King, with an interesting Observational Approach to Populations in Globular Clusters, where discusses the observations on which our understanding of globular clusters lies. Steven Majewski, reviews the Stellar Populations and Formation of the Milky Way, with particular emphasis on the role of globular clusters in tracing stellar populations and unravelling the Galactic history.

The search for the structure of our Universe and our position within it never will cease. As we answer each question, others arise with even greater insistence. And the context of our questions is ever changing. From the mythological background of Babylon, to the mechanical clockwork of Newton, through the opening of our minds to prodigous swarms of distant galaxies, to the mathematical models of general relativity and gravitational clustering within them, each new context inspires new questions. Nor is there reason to suppose that the present context will bring this search to a close.

Throughout the roughhewn matrix of our understanding, dark matter weaves threads of uncertainty. Its amount governs the flight of the galaxies and the fate of the Universe. Many models undertake to confine it to various distributions and forms. So far, dark matter has resisted all but gravitational attempts at detection, leaving the models to flicker and shift in the ebb and flow of theoretical fashion.

Nor do observations always provide simple truths. Most are so riddled with selection and filtered with theory that their interpretation is seldom straightforward. Simple ideas like filaments and voids, walls and clusters, become much more complex when closely examined. Their simple grammar often remains suitable mainly for slogans. All good observers know this in their bones. Results, regardless, can still be astounding.

The history of understanding structure in our Universe is older than the story of Odysseus, and has as many twists and turns. Few of these paths remain familiar to most astronomers today, so in the early chapters I have simply collected some essential developments along the way. They are not without surprises. One of which is that many ideas now thought to be novel have really been known for tens, or hundreds, of years. Often they were no more than speculations, but sometimes they captured reality's core.

The past is our future frontier. Distribution functions at high redshift have not yet been observed. Therefore this chapter will be very short.

Here and there we have glimpses of how galaxy clustering may have developed. These are from observations of two-particle correlations, Lyman alpha clouds, merging protogalaxies, and rich clusters, all at great distances. Eventually, when the halfdozen or so high-redshift catalogs now being started accumulate complete and well-chosen samples, they will yield up the distribution functions of the past. Insofar as these agree with the GQED, their evolution is represented by the changing of b. Thus there is time for genuine predictions, such as those of (30.12) and (30.19)–(30.20) shown in Figure 30.5. These catalogs will also test the importance of merging, which would alter N's conservation (see Chapter 36).

At high redshifts the value of b for gravitational quasi-equilibrium evolution depends quite strongly on Ω0. igures 30.5, 31.12, and 31.13 indicate that as we look further into the past, b will decrease more slowly for lower Ω0. This is essentially because the clustering pattern is “frozen” at higher redshifts for lower Ω0. Equivalently, for higher Ω0 most of the evolution occurs more recently. Zhan (1989) and Saslaw and Edgar (1999) give useful diagrams to show how this can help determine Ω0 and the redshift at which galaxies start clustering.

The total mass of the globular cluster system of our Galaxy makes only ∼10−3 the mass of the Galactic disk. It contains, however, ∼20% of all known low-mass binary X-ray sources, about one half of all binary pulsars, and more than a half of the millisecond pulsars in the Galaxy. Close binary systems containing neutron stars should thus form much more easily in the dense stellar environment of globular clusters than elsewhere in the Galaxy. In these lectures we first review the formation mechanism of neutron stars. Then, we present the evolutionary scenarios leading to the formation of binary X-ray sources and binary and millisecond pulsars in the Galactic disk and the Galactic bulge. We later discuss the specific mechanisms to form neutron star binaries in globular clusters. We end by discussing the open issues concerning the origin and evolution of X–ray sources and millisecond pulsars in globular clusters, and their relationship with the structure, dynamics and evolution of the clusters themselves.

Low–mass binary X–ray sources and millisecond pulsars

An early, unexpected result of X–ray astronomy was the discovery of several bright X–ray sources in globular clusters. Later on, searches for radio pulsars have produced many detections, especially of short period pulsars. We begin these lectures with a very schematic presentation of those two kinds of objects.

X–ray binaries

Luminous Galactic binary X–ray sources provided the first evidence of neutron star binaries, that is binary star systems containing neutron stars (Giacconi et al. 1971; Lewin et al. 1971; Schreier et al. 1972; Tananbaum et al. 1972).

Current theoretical predictions concerning the evolution of old, metal-poor stars in galactic globulars are revisited in the light of recent improvements in the input physics. After a short introduction, the role of fundamental physics in constraining stellar structure all along the various evolutionary phases is shortly recalled, together with the additional role played by the evaluation of some macroscopic mechanisms, like convection and diffusion (Sect. 2). Theoretical predictions concerning the CM-diagram location of the best-known evolutionary phases are discussed in some detail, with particular regard to the existing uncertainties in modeling stellar structure as well as in handling observational data (Sect. 3). This discussion is thus extended to the faint stars recently revealed by HST, either at the faint end of the MS or along the WD cooling sequence (Sect. 4). Additional theoretical constraints given by the pulsational properties of RR Lyrae pulsators are recalled (Sect. 5) and the case of extragalactic globulars in the Local Group are briefly discussed (Sect. 6). Some general and methodological considerations close the paper.

The CM diagram: an introduction

The birth of modern physics dates back to the time when Galileo Galilei stated that any attempt to understand the world around us must—first of all—save the phenomena (“salvare i fenomeni”). In modern words, we say that physics is studying relations between observable quantities, so that the identification of suitable “observables” is a first-priority step in any physical investigation.

Hundreds of examples of two-point correlations have been published for simulated and observed systems. Here we will stick to the basics. First we see how point masses, each representing a galaxy, can start with a Poisson distribution and then correlate gravitationally for different initial velocity dispersions in universes with different critical densities Ω0. Then I summarize some effects of incorporating a range of galaxy masses, and of other initial distributions, and of intergalactic dark matter.

Originally, astronomers hoped that these sorts of results could be compared directly with observations to read the value of Ω0 off the sky. But it was not to be. Too many ambiguities and combinations of conditions gave similar results for ξ2(r). Current observations are not yet sensitive enough to distinguish among all views of the past. Here I describe just some modern examples of the observed form of ξ2 and consider how they may differ for different types of galaxies.

Simulations

Starting a simulation from a Poisson distribution has the attractive feature of starting with minimum structural information. The initial power spectrum (14.35) has n = 0 and thus there is equal power on all scales: a democratic beginning. Then we can watch how pure gravitational interaction builds up different structures. This approach also helps isolate the effects of more complex initial conditions and processes.

First, we consider a related set of simulations with N = 4,000, values of Ω0 = 1, 0.1, and 0.01, and a Poisson initial spatial distribution.

To the denizens of Iolanthe, we can only say with wonder that their motions gravitational are very much more rational, and easier to understand. Not individually, but statistically. To test the special function (29.4) for many-body motions in the context of cosmology we return to simulations in Sections 31.1–31.4.

Figures 15.5 and 15.6 in Section 15.3 illustrated some results of these simulations, which we now examine in more systematic detail. As in Section 31, we start with the simplest Ω0 = 1 case and identical masses. Then we explore the effects of smaller Ω0 and of components with different masses. Unlike the spatial distributions, velocity distribution functions are just beginning to be computed for experiments with dark matter and non-Poisson initial conditions. This is partly because the definition of which particles constitute a galaxy is still unsettled for such cases and partly because observations of f(v) for representative samples are just starting to be analyzed. Both these situations should improve. Then velocity distribution functions will become very valuable because they are more sensitive than the spatial distributions to some of the basic cosmological parameters.

Figure 32.1 shows the velocity distribution functions at four different expansion factors for the 4,000-body, Ω0 = 1, initially cold Poisson simulations with particles of identical masses.

Quick and capacious computers, increasing realization of the importance of largescale structure, and the first glimpses into related many-body physics all combined to change our understanding of galaxy clustering in the early 1970s. So with our historical perspective concluded (though never complete) we now change our approach and describe selected ways to characterize the galaxy distribution. With the large, automatically analyzed catalogs now available, there is no lack of positional data. Successful new observational techniques are also providing many galaxy redshifts, which are being refined into peculiar velocities relative to the general expansion. Nor is there any lack of statistical techniques for analyzing the data. Dozens of quantitative descriptions, many based on analogies in subjects ranging from archaeology to zoology, have been proposed. The main problem is to select those which give most insight into the physical causes of the structure we see. In the next chapters, I sketch several examples, their strengths and weaknesses, and some of their accomplishments. It helps provide a perspective for the two descriptions that will dominate subsequent chapters: correlation functions and distribution functions.

Globular cluster systems represent only a small fraction of the total stellar mass of galaxy halos, but provide unique tracers which can be used to address models of galaxy formation. Several “case studies” of individually important galaxies are presented, in which we look at the characteristics of their globular clusters including the metallicity distributions, specific frequencies, luminosity (mass) distributions, and kinematics. Among these galaxies are the Milky Way, the nearby giant elliptical NGC 5128, the Virgo ellipticals NGC 4472 and M87, and the supergiant cD galaxies at the centers of rich clusters. In each case the possible roles of mergers, small-satellite accretions, and in situ formation in the growth of the galaxy are discussed. We also briefly touch on the connection between the globular clusters and the much more numerous field-halo stars. We conclude that in all formation scenarios, the presence or absence of gas at any stage of the galaxy's evolution plays a crucial role in determining the total cluster population, the number of distinguishable subpopulations, and the metallicity distribution of the clusters.

The analysis of globular cluster systems (GCSs) in other galaxies is starting to fulfil its long-held promise of informing us about galaxy formation in ways that are unique. The more we learn about GCSs, the more we realize that their role in galaxy formation is an intricate and varied process – yet with common themes that apply particularly to the old-halo population that is found in every type of galaxy.

In the past, there has been a widely held mythology that thermodynamics and gravity are incompatible. The main arguments for that view were threefold. First, thermodynamics applies to equilibrium systems. But self-gravitating systems continually evolve toward more singular states, so they are never in equilibrium. Second, to obtain a thermodynamic or statistical mechanical description it must be possible to calculate the partition function for the relevant ensemble, as in (23.60). But self-gravitating systems contain states where two objects can move arbitrarily close and contribute infinite negative gravitational energy, making the partition function diverge. Third, the fundamental parameters of thermodynamics must be extensive quantities. But self-gravitating systems cannot localize their potential energy in an isolated cell; it belongs to the whole system.

All three of these arguments have a common basis in the long-range nature of the gravitational force and the fact that it does not saturate. By contrast, in the electrostatic case of a plasma, although the Coulomb forces are also long range, the positive and negative charges effectively cancel on scales larger than a Debye sphere where the plasma is essentially neutral, and its net interaction energy is zero. So one can describe plasmas thermodynamically (e.g., Landau & Lifshitz, 1969).